Expenditure function

In microeconomics, the expenditure function gives the minimum amount of money an individual needs to spend to achieve some level of utility, given a utility function and the prices of the available goods. Formally, if there is a utility function that describes preferences over n commodities, the expenditure function says what amount of money is needed to achieve a utility if the n prices are given by the price vector . This function is defined by where is the set of all bundles that give utility at least as good as . Expressed equivalently, the individual minimizes expenditure subject to the minimal utility constraint that giving optimal quantities to consume of the various goods as as function of and the prices; then the expenditure function is (Properties of the Expenditure Function) Suppose u is a continuous utility function representing a locally non-satiated preference relation o on Rn +. Then e(p, u) is

1. Homogeneous of degree one in p: for all and >0,
2. Continuous in and
3. Nondecreasing in and strictly increasing in provided
4. Concave in
5. If the utility function is strictly quasi-concave, there is the Shephard's lemma Proof (1) As in the above proposition, note that (2) Continue on the domain : (3) Let and suppose . Then , and . It follows immediately that . For the second statement , suppose to the contrary that for some , Than, for some , , which contradicts the "no excess utility" conclusion of the previous proposition (4)Let and suppose . Then, and , so . (5) The expenditure function is the inverse of the indirect utility function when the prices are kept constant. I.e, for every price vector and income level : There is a duality relationship between expenditure function and utility function. If given a specific regular quasi-concave utility function, the corresponding price is homogeneous, and the utility is monotonically increasing expenditure function, conversely, the given price is homogeneous, and the utility is monotonically increasing expenditure function will generate the regular quasi-concave utility function.